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Theorem llynlly 21757
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 21752 . 2 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 llyi 21754 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 simpl1 1182 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5 simprl 767 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
6 simprr2 1213 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
7 opnneip 21399 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
84, 5, 6, 7syl3anc 1362 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9 simprr1 1212 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝑥)
10 selpw 4454 . . . . . . 7 (𝑢 ∈ 𝒫 𝑥𝑢𝑥)
119, 10sylibr 235 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
128, 11elind 4087 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13 simprr3 1214 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
142, 12, 13reximssdv 3236 . . . 4 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
15143expb 1111 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
1615ralrimivva 3156 . 2 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
17 isnlly 21749 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
181, 16, 17sylanbrc 583 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1078  wcel 2079  wral 3103  wrex 3104  cin 3853  wss 3854  𝒫 cpw 4447  {csn 4466  cfv 6217  (class class class)co 7007  t crest 16511  Topctop 21173  neicnei 21377  Locally clly 21744  𝑛-Locally cnlly 21745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-top 21174  df-nei 21378  df-lly 21746  df-nlly 21747
This theorem is referenced by:  llyssnlly  21758  symgtgp  22381
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